Is It Possible to Design a System of Mathematics in Which Division by Zero is Defined?

Question

I have heard that non-Euclidean geometry has been developed, in which the shortest distance between two points is not a straight line, so I was wondering if anyone has come up with a mathematical system in which division by zero is possible.  If so, who did so, and what were the results?

Answer 1

"in which the shortest distance between two points is not a straight line"

There is a habit of popular science presenters to say stuff like this but basically they are misleading you by omitting a lot of information or often using words outside of their everyday meaning, eg space is curved, space is flat.



I am not aware of any system in which division by zero is possible. Really, you would need to redefine division or zero because the issue is very simple; Division is defined as multiplying by the "multiplicative inverse" and there is no multiplicative inverse for zero. That is,



We define division by B as,

A ÷ B = A x B' 

where B x B' = 1



But for B = 0 there is no value for B' such that 0 x B' = 1

Answer 2

Zeno explored Math by testing limits.

The eponymous paradox holds that you can't 

travel from point A to point B because there are

an infinite number of half steps in between.



Set up a simple Algebraic expression: y = x^-1

(See graph below)

As x approaches zero, y increases  without bound

but can't get there because x can always get closer

to zero. As x increases without bound

y approaches zero but can't get there because x can 

always get bigger.

Isn't that close enough to explain why division 

by zero is undefined?

Answer 3

A long time ago, when I was in college, I tried to do it; but eventually the equations I developed collapsed into mathematical paradoxes.



As far as I know this simply doesn't work using any presently accepted branch of math.



Back in 1734, mathematician George Berkeley was able to prove that it is impossible to do so.

Answer 4

Many people have tried, but none were serious math people.  --- Every couple of months someone on this site will try but they have no complete mathematical system most of them want to define division by 0 as either 0 or as infinity, but they have no logic behind their 'claim' and nothing beyond that to make an entire system.  

 . . . If you do such a 'system' then you need to show any possible advantages it would have and any possible applications .. No one ever does those things.  I do not think it is possible.